Let $T: \mathbb{R}^3 \to \mathbb{R}^5$ be a linear transformation such that
$\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} -1 \\ 3 \\ 0 \\ 8 \\ 0 \end{pmatrix}$, $T\begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 8 \\ -1 \\ 0 \\ 2 \\ 0 \end{pmatrix}$, $T\begin{pmatrix} 3 \\ 4 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ -3 \\ 8 \\ -2 \end{pmatrix}$.
Find the standard matrix representation and the $x, y, z$ formula $T\begin{pmatrix} x \\ y \\ z \end{pmatrix}$ of $T$.
